"Preserve current filename in XMP metadata" grayed out on Mac Catalina?

A couple of things to try:

* Right click on the image and purge the cache for the selected file, then see if you can preserve. One can also purge the cache for the entire directory/folder via the Tools menu

* Reset Bridge Preferences:

https://helpx.adobe.com/bridge/kb/troubleshoot-errors-freezes-bridge.html

Outside of Bridge, one can use 3rd party software such as ExifTool to embed this metadata:

Add/Copy filename to Adobe Preserve Original Filename metadata entry (excluding file extension), -Preserving the modified date/time:

exiftool -P -overwrite_original_in_place '-XMP-xmpMM:PreservedFileName<${filename;s/\.[^.]*$//}' -r '/Users/username/Desktop/file to rename.png'

Add/Copy filename to Adobe Preserve Original Filename metadata entry (including file extension):

exiftool -overwrite_original_in_place '-XMP-xmpMM:PreservedFileName<${filename}' -r '/Users/username/Desktop/file to rename.png'

These commands are formatted for the Mac, Windows would use double straight quote/inch marks and OS specific paths to directories or files. Always work on copies of your images until you know that all is working as intended.

Not all file types support XMP metadata.